{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T13:22:46Z","timestamp":1767014566421,"version":"3.48.0"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2025,12]]},"abstract":"Abstract<\/jats:title>\n \n A family of\n r<\/jats:italic>\n distinct sets\n \n \n $$\\{A_1,\\ldots , A_r\\}$$<\/jats:tex-math>\n \n \n {<\/mml:mo>\n \n A<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n A<\/mml:mi>\n r<\/mml:mi>\n <\/mml:msub>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is an\n r<\/jats:italic>\n -sunflower if for all\n \n \n $$1 \\leqslant i < j\\leqslant r$$<\/jats:tex-math>\n \n \n 1<\/mml:mn>\n \u2a7d<\/mml:mo>\n i<\/mml:mi>\n <<\/mml:mo>\n j<\/mml:mi>\n \u2a7d<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$1 \\leqslant i' < j'\\leqslant r$$<\/jats:tex-math>\n \n \n 1<\/mml:mn>\n \u2a7d<\/mml:mo>\n \n i<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <<\/mml:mo>\n \n j<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n \u2a7d<\/mml:mo>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , we have\n \n \n $$A_i\\cap A_j = A_{i'}\\cap A_{j'}$$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n \u2229<\/mml:mo>\n \n A<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \n A<\/mml:mi>\n \n i<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <\/mml:msub>\n \u2229<\/mml:mo>\n \n A<\/mml:mi>\n \n j<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Erd\u0151s and Rado conjectured in 1960 that every family\n \n \n $$\\mathcal {H}$$<\/jats:tex-math>\n \n H<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of\n \n \n $$\\ell $$<\/jats:tex-math>\n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -element sets of size at least\n \n \n $$K(r)^\\ell $$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \n \n (<\/mml:mo>\n r<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2113<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n contains an\n r<\/jats:italic>\n -sunflower, where\n K<\/jats:italic>\n (\n r<\/jats:italic>\n ) is some function that depends only on\n r<\/jats:italic>\n . We prove that if\n \n \n $$\\mathcal {H}$$<\/jats:tex-math>\n \n H<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a family of\n \n \n $$\\ell $$<\/jats:tex-math>\n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -element sets of VC-dimension at most\n d<\/jats:italic>\n and\n \n \n $$|\\mathcal H| > (C r(\\log d+\\log ^*\\ell ))^\\ell $$<\/jats:tex-math>\n \n \n \n |<\/mml:mo>\n H<\/mml:mi>\n |<\/mml:mo>\n ><\/mml:mo>\n <\/mml:mrow>\n \n \n (<\/mml:mo>\n C<\/mml:mi>\n r<\/mml:mi>\n \n (<\/mml:mo>\n log<\/mml:mo>\n d<\/mml:mi>\n +<\/mml:mo>\n \n log<\/mml:mo>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \u2113<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2113<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for some absolute constant\n \n \n $$C > 0$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , then\n \n \n $$\\mathcal {H}$$<\/jats:tex-math>\n \n H<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n contains an\n r<\/jats:italic>\n -sunflower. This improves a recent result of Fox, Pach, and Suk. When\n \n \n $$d=1$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , we obtain a sharp bound, namely that\n \n \n $$|\\mathcal H| > (r-1)^\\ell $$<\/jats:tex-math>\n \n \n \n |<\/mml:mo>\n H<\/mml:mi>\n |<\/mml:mo>\n ><\/mml:mo>\n \n (<\/mml:mo>\n r<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n \u2113<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is sufficient. Along the way, we establish a strengthening of the Kahn\u2013Kalai conjecture for set families of bounded VC-dimension, which is of independent interest.\n <\/jats:p>","DOI":"10.1007\/s00493-025-00186-8","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T11:26:29Z","timestamp":1762514789000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sunflowers in Set Systems with Small VC-Dimension"],"prefix":"10.1007","volume":"45","author":[{"given":"J\u00f3zsef","family":"Balogh","sequence":"first","affiliation":[]},{"given":"Anton","family":"Bernshteyn","sequence":"additional","affiliation":[]},{"given":"Michelle","family":"Delcourt","sequence":"additional","affiliation":[]},{"given":"Asaf","family":"Ferber","sequence":"additional","affiliation":[]},{"given":"Huy Tuan","family":"Pham","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"186_CR1","unstructured":"Allen, P., B\u00f6ttcher, J., Kohayakawa, Y., Neve, M.: Robustness of the Sauer\u2013Spencer Theorem. arXiv:2507.03676 (preprint). 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